Salut! ⚛
_________
Determinați, în fiecare caz, unghiul vectorilor.
Fie u(x₁, y₁) și v(x₂, y₂) și ∝ = m(∡u, v)
[tex]a)[/tex] [tex]u(2, -1)[/tex][tex],[/tex] [tex]v(1,2)[/tex]
[tex]cos\alpha =\frac{x_{1} *x_{2}+y_{1} *y_{2} }{\sqrt{x_{1}^{2} +y_{1} ^{2} }*\sqrt{x_{2}^{2} +y_{2}^{2} } }[/tex]
[tex]cos\alpha =\frac{2*1+(-1)*2}{\sqrt{4+1}*\sqrt{1+4} } => cos\alpha =\frac{2-2}{\sqrt{5} *\sqrt{5} } =\frac{0}{5} =0[/tex]
[tex]cos\alpha =0 => \alpha =90 ^{0}[/tex]
[tex]b)[/tex] [tex]u(4, -2),[/tex] [tex]v(1,2)[/tex]
[tex]cos\alpha =\frac{x_{1} *x_{2}+y_{1} *y_{2} }{\sqrt{x_{1}^{2} +y_{1} ^{2} }*\sqrt{x_{2}^{2} +y_{2}^{2} } }[/tex]
[tex]cos\alpha =\frac{4*1+(-2)*2}{\sqrt{16+4} *\sqrt{1+4} } =\frac{4-4}{\sqrt{20}*\sqrt{5} } =\frac{0}{\sqrt{100} } =0[/tex]
[tex]cos\alpha =0 => \alpha =90 ^{0}[/tex]
[tex]f)[/tex] [tex]u(4, -3),[/tex] [tex]v(-3, 4)[/tex]
[tex]cos\alpha =\frac{x_{1} *x_{2}+y_{1} *y_{2} }{\sqrt{x_{1}^{2} +y_{1} ^{2} }*\sqrt{x_{2}^{2} +y_{2}^{2} } }[/tex]
[tex]cos\alpha =\frac{4*(-3)+(-3)*4}{\sqrt{16+9} *\sqrt{9+16} } =\frac{-12-12}{\sqrt{25} *\sqrt{25} } =-\frac{24}{25}[/tex]
[tex]cos(2\pi -\alpha )=-\frac{24}{25}=> 2\pi -\alpha =arccos(-\frac{24}{25} ) => 2\pi -\alpha =\pi -arccos(\frac{24}{25} )[/tex] [tex]{\mathbf{(1)}[/tex]
[tex]cos(\alpha )=-\frac{24}{25} => \alpha =\pi -arccos(\frac{24}{25} )[/tex] [tex]{\mathbf{(2)}[/tex]
[tex]Din[/tex] [tex](1)[/tex] [tex]si[/tex] [tex](2)[/tex] [tex]=>[/tex] [tex]2\pi -\alpha =\alpha => 2\alpha =2\pi => \alpha =\pi => \alpha =180^{0}[/tex]
#copaceibrainly
